Analysis of the Effect of Uncertain Average Winds on Cruise Fuel Load Rafael V´ azquez Dami´ an Rivas University of Seville, Spain 5th SESAR Innovation Days, Bologna, December 2015
1 Propagation of Uncertainties Introduction Problem Statement 2 Methods and Results Generalized Polynomial Chaos (GPC) Probability density function Transformation Method (PTM) Results 3 Conclusions & Future Work
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Introduction ATM: a very complex system with a large number of heterogeneous components 3 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Uncertainties in ATM ATM operations and in particular aircraft trajectories are subject to many uncertainties. Sources of uncertainty include: wind and severe weather navigational errors aircraft performance inaccuracies errors in the FMS... 4 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Uncertainties in ATM ATM operations and in particular aircraft trajectories are subject to many uncertainties. Sources of uncertainty include: wind and severe weather navigational errors aircraft performance inaccuracies errors in the FMS... The analysis of the impact of uncertainties in aircraft trajectories and its propagation through the flight segments is of great interest Study sensitivity of the system to lack of precise data / measurement errors Aid in the design of a more robust ATM system 4 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Contribution of this work Analyze promising tools to study uncertainty propagation, comparing with analytical results 5 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Contribution of this work Analyze promising tools to study uncertainty propagation, comparing with analytical results To obtain analytical results, a simplified case has been considered: Analysis of the Effect of Uncertain Average Winds on Cruise Fuel Load 5 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Contribution of this work Analyze promising tools to study uncertainty propagation, comparing with analytical results To obtain analytical results, a simplified case has been considered: Analysis of the Effect of Uncertain Average Winds on Cruise Fuel Load Wind is the main source of uncertainty in trajectory prediction Mass evolution largely determines fuel consumption and thus flight cost → study uncertainty propagation through mass dynamics Cruise uncertainties have a large impact on the overall flight Mass evolution in cruise flight: single nonlinear equation, analytically solvable. Some interesting conclusions can be derived. 5 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Trajectory uncertainty propagation Methods to study trajectory uncertainty propagation fall in two categories parametric: the statistical moments (mean, variance) are propagated non-parametric: the full probability density function is evolved In this presentation both type of methods are considered Monte-Carlo methods: innumerable works have used these, however they are not very precise and computationally very intensive ☛ ✟ More methods, more results : take a look at the paper or visit http://complexworld.eu/wiki/Uncertainty_propagation ✡ ✠ 6 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Mass dynamics in cruise flight Assumptions: Cruise flight Other typical flight mechanics assumptions (symmetric flight, parabolic polar, etc) m = − ( A + Bm 2 ) Mass equation can be written as ˙ Horizontal distance equation: x = V + w ˙ where x is the horizontal distance, and w average wind speed. Combining mass and distance equations: dx = − A + Bm 2 dm V + w The solution to this equation is used to determine total fuel consumed for a given distance 7 / 23
Propagation of Uncertainties Introduction Methods and Results Problem Statement Conclusions & Future Work Uncertainty in average wind f w 1 2 ± w ± w w w Consider that the average wind w is not known a priori, but rather a random variable Objective of this work: ✞ ☎ Find how the uncertainty in average wind affects the fuel cost ✝ ✆ Easy probabilistic model for w : the Uniform distribution w is the mean and δ w the width ¯ 8 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Propagation of uncertainty increasing time 9 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Methods The following tools are used to study uncertainty propagation of initial mass uncertainty: Generalized Polynomial Chaos (GPC) A numerical method to propagate distribution functions: Probability density function Transformation Method (PTM) Polynomial chaos is fast and precise (compared with Monte Carlo methods). However can only be used to obtain statistical properties such as mean or variance Distribution functions contain all information about the random variable and thus are very useful to have. 10 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Introduction to GPC I Representation of a random process as a Fourier-type series, with time-dependent coefficients introduced by Norbert Wiener in 1938 Orthogonal polynomials as GPC basis functions 11 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Introduction to GPC I Representation of a random process as a Fourier-type series, with time-dependent coefficients introduced by Norbert Wiener in 1938 Orthogonal polynomials as GPC basis functions What polynomials are best? Choose orthogonal polynomials with respect to mathematical expectation. Then the convergence of the series is exponential Orthogonality: given two polynomials φ i and φ j , then E [ φ i φ j ] = 0, if i � = j For the uniform distribution: Legendre polynomials 11 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Introduction to GPC II Thus expand the mass of the aircraft P � m ( t ) = h i ( t ) L i i =0 The coefficients h i ( t ) verify ODEs which are found from mass differential equation (details in the paper) P is the order of the approximation Once the coefficients h i are computed, calculate mean and variance The advantage of the GPC method is that a small or moderate value of P is enough to get good results, thus resulting in a computationally much less intensive method than Monte-Carlo simulations 12 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results GPC convergence × 10 4 1500 8 Convergence was obtained with 7.5 1000 7 h 0 (x) h 2 (x) P = 4 (see the aircraft 6.5 500 6 parameters in the paper) 5.5 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 × 10 6 x × 10 6 x Note how each mode is 0 0 approximately one order of -50 -2000 magnitude smaller than the h 1 (x) -100 h 3 (x) -4000 -150 previous one -6000 -200 -8000 -250 Additional modes do not provide 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 × 10 6 x × 10 6 x significant changes in the values 40 of the mean or typical deviation 30 h 4 (x) 20 10 0 0 0.5 1 1.5 2 2.5 × 10 6 x 13 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Distribution functions of transformed variables The basic idea of PTM is based on the following basic result from statistics: Result Given a random variable x with probability density function f x ( x ), if a new random variable is defined as y = g ( x ), then f y ( y ) is f y ( y ) = f x ( g − 1 ( y )) | g ′ ( g − 1 ( y )) | 14 / 23
Propagation of Uncertainties Generalized Polynomial Chaos (GPC) Methods and Results Probability density function Transformation Method (PTM) Conclusions & Future Work Results Distribution functions of transformed variables The basic idea of PTM is based on the following basic result from statistics: Result Given a random variable x with probability density function f x ( x ), if a new random variable is defined as y = g ( x ), then f y ( y ) is f y ( y ) = f x ( g − 1 ( y )) | g ′ ( g − 1 ( y )) | Denoting m F = g ( w ) as the fuel mass obtained by solving the differential equation for each wind w : f m F ( m F ) = f w ( g − 1 ( m F )) | g ′ ( g − 1 ( m F )) | 14 / 23
Recommend
More recommend
